#### Unsteady-State Energy Balances on Tanks: Screencasts

When the valve on an adiabatic tank is opened and some of the ideal gas is released, the temperature and the amount of the remaining gas is calculated as a function of the final pressure in the tank.

We suggest that after watching this screencast, you list the important points as a way to increase retention.

Uses an interactive simulation to describe the impact of the state of the feed to a distillation column on the liquid and vapor flow rates in the column.

We suggest that after watching this screencast, you list the important points as a way to increase retention.

##### Important Equations:

For an ideal gas:

$\Delta U = C_V\Delta T; \hspace{3mm} \Delta H = C_P\Delta T$

where $$\Delta U$$ is the change in internal energy per mole
$$C_V$$ is the constant volume molar heat capacity
$$C_P$$ is the constant pressure molar heat capacity
$$\Delta H$$ is the change in enthapy per mole

Adiabatic reversible expansion for an ideal gas is applied to the expanding gas that remains in the tank:

$\frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right) ^\frac{R}{C_P}$

where $$P_1$$ and $$T_1$$ are the initial pressure and temperature
$$P_2$$ and $$T_2$$ are the final pressure and temperature. The temperatures must be absolute (K), and $$R$$ is the ideal gas constant.

Ideal gas law for opening tank valve to release some gas and then closing valve:

$P_1V = n_1RT_1 \hspace{5mm} P_2V = n_2RT_2$

where $$V$$ is the tank volume
$$n_1$$ is the initial number of moles in the tank
$$n_2$$ is the final number of moles in the tank

Energy balance for adiabatic evaporation of a fraction of liquid into a vacuum:

$\Delta U^t = 0$

where $$\Delta U^t$$ is the total energy change of the system (liquid and vapor). The container is assumed to have no heat capacity

$m_f ^{vap} U_f ^{vap} + m_f ^{liq} U_f ^{liq} = m_i U_i ^{liq}$

$m_i = m_f ^{vap} + m_f ^{liq}$

where $$m_i$$ is the initial mass of liquid
$$m_f ^{vap}$$ is the final mass of vapor
$$m_f ^{liq}$$ is the final mass of liquid
$$U_f ^{vap}$$ is the internal energy (per g or per mol) of the vapor at equilibrium
$$U_f ^{liq}$$ is the internal energy (per g or per mol) of the liquid at equilibrium
$$U_i ^{liq}$$ is the internal energy (per g or per mole) of the initial liquid

$U^{liq} = U_{ref} ^{liq} + C_V ^{liq} (T – T_{ref})$

where $$U^{liq}$$ is the internal energy of liquid at temperature $$T$$
$$U_{ref} ^{liq}$$ is the internal energy of liquid at the reference temperature $$T_{ref}$$
$$C_V ^{liq}$$ is the constant-volume heat capacity of the liquid, which is essentially the same as $$C_P ^{liq}$$.

$U^{vap} = U_{ref} ^{liq} + \Delta U_{ref} ^{vap} + C_V ^{vap} (T – T_{ref})$

where $$U^{vap}$$ is the internal energy of the vapor at temperature $$T$$
$$\Delta U_{ref} ^{vap}$$ is the change in internal energy for vaporization at $$T_{ref}$$
$$C_V ^{vap}$$ is the constant-volume heat capacity of the vapor

At equilibrium, $$T_{liquid} = T_{vapor}$$

Gas phase is assumed to be an ideal gas.

Antoine equation:

$lnP^{sat} = A – \frac{B}{T+C}$

where $$P^{sat}$$ is the saturation pressure at temperature $$T$$
$$A$$, $$B$$, and $$C$$ are constants